The Complex Asymptotic Theory of a Fourth Order Differential Equation of Hydrodynamics

for large values of the parameter X. Here N[y] and M[y] are linear differential expressions of the function y = y(x), of order n and m, respectively. The essential feature of the problem lies again in the assumption that n > m. In [9] the main question was under what conditions a solution of (1) with prescribed boundary values will tend to a solution of M[y] = 0, as X tends to infinity. In the present investigation we shall also discuss, as an application, certain boundary value problems, but our principal task consists here in finding a fundamental system of solutions of (1) whose asymptotic properties are known in a given domain of the complex x-plane. In addition to this shift of emphasis our problem differs in three respects from the one treated in [9]. We shall assume that 1. The independent variable x ranges over a domain of the complex plane. 2. The leading coefficient of the differential expression M[y] has a zero of the first order in the domain under consideration, whereas the leading coefficient of N[y] does not vanish there. 3. We limit ourselves to the case that N[y] is of order n = 4, and M[y] is of order m = 2. There are mathematical as well as physical reasons for studying this problem. Mathematically, an essential difference between the solutions of the full problem (1) and those of the reduced problem M[y] = 0 should be noted. The former are regular singlevalued functions in the neighborhood of the point where the leading coefficient of M[y] vanishes. But the solutions of M[y] = 0 will, in general, be multivalued, since they have a branch point there. Such multivalued functions can obviously not be uniform limits of solutions of (1) in a